Decompose the <optical tensor> on the two-dimensional screen space as
$$
\widehat B_{ab}
=\frac12\theta P_{ab}
+\widehat\sigma_{ab}
+\widehat\omega_{ab}.
$$
The <null expansion>, <null shear>, and <null twist> are respectively
$$
\boxed{\theta=P^{ab}\widehat B_{ab}},
\qquad
\boxed{\widehat\sigma_{ab}
=\widehat B_{(ab)}-\frac12\theta P_{ab}},
\qquad
\boxed{\widehat\omega_{ab}=\widehat B_{[ab]}}.
$$
The shear is symmetric and trace-free, while the twist is antisymmetric.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 1 / a / iii / Solution

{title2=$\widehat\sigma_{ab}$}

The null shear is the symmetric trace-free part $\widehat\sigma_{ab}=\widehat B_{(ab)}-\tfrac12\theta P_{ab}$. It changes the shape of a transverse beam while preserving its area to first order.


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